The conjecture that blowup Ramsey numbers must depend on the base graph

Let $G \overset{r}{\longrightarrow} H$ mean that every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$, and let $\mathrm{B}(G \overset{r}{\longrightarrow} H;t)$ be the minimum $n$ such that every $r$-coloring of $G[n]$ contains a monochromatic canonical copy of $H[t]$. Dependence-on-$G$ conjecture. There exists a graph $H$ and integers $r,t\geq 2$ such that there are graphs $G_1,G_2,\ldots$ with $G_i \overset{r}{\longrightarrow} H$ for every $i$ and

$$\sup_i \mathrm{B}(G_i \overset{r}{\longrightarrow} H;t)=\infty.$$

This asserts that, although the exponential constant can be made independent of $G$, the blowup Ramsey number itself cannot in general be bounded independently of $G$. The source reports counterexamples for certain $H$ and $r$, so this conjecture is refuted.